Simultaneous proximinality in \(L^∞(μ,X)\) - Abu-Sirhan, Eyad (Rev. Anal. Numér. Théor. Approx., 2013)
Let \(X\) be a Banach space and \(G\) be a closed subspace of \(X\). Let us denote by \(L^∞( μ,X) \) the Banach space of all \(X\)-valued essentially bounded functions on a \(σ\)-finite complete measure space \(( Ω,Σ,μ) .\) In this paper we show that if \(G\) is separable, then \(L^∞( μ,G) \) is simultaneously proximinal in \(L^∞( μ,X) \) if and only if \(G\) is simultaneously proximinal in \(X.\)